% $Id: paper.tex 204 2009-04-20 01:59:36Z jsibert $
%
% Author: David Fournier
% Copyright (c) 2008 Regents of the University of California
%

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\begin{document}
\def\ABCj{ABC_j }
\def\mABCj{$\ABCj$ }
\def\Bj{B_j }
\def\mBj{$\Bj$ }
\def\mj{$j$  }
\def\mb{$b$  }
\def\mk{$k$  }
\def\Fabc{F_{ABC} }
\def\mFabc{$F_{ABC}$ }
\def\thetaj{\theta_j }
\def\mthetaj{$\theta_j$ }
\def\TAC{TAC }
\def\TACj{TAC_j }
\def\mTAC{$\TAC$ }
\def\OY{OY }
\def\mOY{$\OY$ }
\def\Yj{Y_j }
\def\mYj{$\Yj$ }
\def\Pj{P_j }
\def\mPj{$\Pj$ }
\def\bjk{b_{jk} }
\def\mbjk{$b_{jk}$ }
\tableofcontents
\newpage
\openup 10truept
\chapter{The Model}
\section{Introduction}

For a given year let \mABCj be the ``Acceptible Biological Catch''
for species~$j$. \hbox{$\ABCj=f(B_j,\theta_j)$} where \mBj represents the
biomass of species \mj  and \mthetaj is avector of parameters
describing the age-specific schedules affecting estimates of 
maximum \mFabc harvest rates. Let \mTAC (Total Allowable Catch)
have constraints such that $\TAC\le \ABCj$ and $\sum_j\TACj\le \OY$
where \mOY is the maximum allowable catch of all species combined.
Let \mYj be the yield of species \mj where $\Yj=f(\TACj,\bjk,\Pj)$,
with \mbjk representing the relative harvest rate of species \mj
in fishery \mk and \mPj is the (implied) revenue earned by catching species
\mj. The diagonals for the matrix \mb represent the
allowance of each species catch in the directed fishery for that species
whereas the off diagonal elements represent the bycatch rates.
The problem is  to maximize the revenue $R=\sum_j \Pj\Yj$
subject to the constraints outlined above.

\section{An example}

Consider the following example for three species. Let
$ABC=\big(1.4\ 1.0\ 0.01\big)$, $\OY=2.0$, and
$P=\big(0.4\ 0.2\ 0.0\big)$. Further we note that species~1 and 2 
are ``target'' species, and species~3 represents a ``Prohibited
species'' such that
\begin{displaymath}
b =\left(\matrix{1.0&0.25&0.0\cr
      0.11&1.0&0.0\cr
      0.14&1.0&0.0\cr}\right )
\end{displaymath}




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